#9223960

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4341 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(9,10,14,18)(11,19,21,15)(12,16,22,20)(13,17,24,23)(26,27)(29,37)(31,33)(35,38);(1,2)(3,4)(5,7)(6,8)(9,11,14,21)(10,15,18,19)(12,17,22,23)(13,20,24,16)(25,26,30,27)(28,33,36,31)(29,32,37,34)(35,40,38,39);(1,3)(2,6)(4,7)(5,8)(25,26,30,27)(28,31,36,33)(29,32,37,34)(35,39,38,40);(1,4,5,6)(2,3,7,8)(9,10)(11,15)(12,20)(13,23)(14,18)(16,22)(17,24)(19,21)(25,27,30,26)(28,31,36,33)(29,32,37,34)(35,40,38,39);(1,3)(2,4)(5,8)(6,7)(9,12)(10,16)(11,17)(13,15)(14,22)(18,20)(19,24)(21,23)(25,28,30,36)(26,31,27,33)(29,38,37,35)(32,40,34,39);(1,2)(3,6)(4,8)(5,7)(9,13)(10,17)(11,20)(12,19)(14,24)(15,22)(16,21)(18,23)(25,29)(26,32)(27,34)(28,35)(30,37)(31,39)(33,40)(36,38)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=hi,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9131156);